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General edge asymptotics of solutions of second-order elliptic boundary value problems I

Published online by Cambridge University Press:  14 November 2011

Martin Costabel  and
Monique Dauge
Martin Costabel
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
Monique Dauge
Affiliation:
Département de Mathématiques, Université de Nantes, 2, rue de la Houssinière, 44072 Nantes Cedex 03, France
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Synopsis

This is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 109 - 155
Copyright
Copyright © Royal Society of Edinburgh 1993

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